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Full text of "Treatise On Analysis Vol-Ii"

21    PRODUCT OF MEASURES        235

system in ^^(l.X) (show that they are all orthogonal to the functions co$2k7rt
for k integral and ^>0). Show that /•„(/) = sgn(sin 2n7rt).

(c)    For each x =(*„)„Ł 0 in X, put u(x) = (xn + l)n^0. Show that the measure p, is
invariant with respect to u and that u is ergodic with respect to //, (use (a) and Problem
5(c) of Section 13.12). Deduce that

lim -(ri(/)+ •••+rn(/)) = 0

n-voo n

almost everywhere with respect to Lebesgue measure ("Borel-Cantelli theorem").
(Observe that rn(\ - f) = -rB(0 for 0 <i t <[ J.)

(d)    For each finite strictly increasing sequence (nt)1Łt^k of integers ^ 1, show that

(e)   For each finite sequence (ak)i$k^n of complex numbers, and each real number
p > 0, show that

r

P/2

("Khintchine's inequality"). (First consider the case where/? = 2/z, with h an integer
2:1, by using (d). For 2h — 2<p<:2h, use problem 12(e) of Section 13.11.)
(f)   With the notation of (e), show that

(n          \l/2            /•!          n

fc.i        /         ^        Jo       *=1

(use (e) for p = 4, and Holder's inequality for p = | and q — 3).

11. (a) The hypotheses are those of Section 13.17, Problem 2, and in addition the
functions /„ are assumed to be real. Let xh-+j(x) be a ju-measurable mapping of X
into the set I = {1, 2,..., /?}, and let k\—*w(k) be an increasing function on I with
values >0. Then the mapping (s9 w)h~>Ky(S)(5<, u)lw(j(s)) of X x X into R is p, Ž /it-
measurable. For each /^-measurable subset A of X, show that

where h(s, t) = inf(j(s)J(t)). (Express the square of the integral on the left as a
double integral, so as to reduce the problem to the evaluation of a triple integral
over A x A x X, and use the fact that the /„ are orthogonal.) Show next that

r      |K*(M,(j, 01 J

(split up A x A into two measurable sets, defined respectively by j(s) <j(t) and
J($) Sy'(') for (s, 0 e A x A and use the fact that Kn(s, r) = Kn(f, s)).
(b) Suppose that there exists an increasing sequence n \-*w(ri) of numbers >0 and
a /x-measurable subset A of X such that |HnC?)| ^ c • yv(n) for all ^ e A and all n ^ 1,
where c> 0 is a constant. Show that, for each function ge JS?Ł(X, /A), the sequence
(sn(g)(t))lw(ri) is bounded for almost all t e A, (Consider the increasing sequence of
u-integrable functions vn(t) — sup (j*(^)(/))/w(A:), and show that the sequence off X let un be the measure for which ju,n(0) = /^n(l) = i, and let a = Ž un be the